Algebraic Equation
An algebraic equation is an equation in the form:
P = 0
Where P is a polynomial.

For example, x + 8 = 0 is an algebraic equation, where x + 8 is a polynomial. Hence, it is also called a polynomial equation.

An algebraic equation is always a balanced equation that includes variables, coefficients, and constants.

Consider an equation 1+1 = 2.

It is balanced as both sides have the same value. To avoid committing an error that tips the equation out of balance, make sure that any change on one side of the equation is reciprocated on the other side. For example, if you want to add a number 5 to one side of the equation you will have to add the same 5 to the other side of the equation i.e.

1 + 1 = 2

1 + 1 + 5 = 2 + 5

The same goes for subtraction, multiplication, and division. As long as you do the same thing to both sides of the equation it will remain balanced.
The solution of an algebraic equation is the process of finding a number or set of numbers that, if substituted for the variables in the equation, reduce it to an identity. Such a number is called a root of the equation.

Transcendental Equation
A transcendental equation is an equation into which transcendental functions (such as exponential, logarithmic, trigonometric, or inverse trigonometric) of one of the variables (s) have been solved for. Transcendental equations do not have closed-form solutions.

Transcendental equations examples includes: x=e−x,x=cosx,2x=x²

1. Find dy/dx for the function y = In(tan x + sec x)
   

Solution:

dy/dx = x² (1/4x. 4) + In (4x). 2x

= x + 2x In ( 4x)

= x( 1 + 2 In (4x))